Suppose in space some group or other, the principal group for instance, be given. Let us then select a single configuration, say a point, or a straight line, or even an ellipsoid, etc., and apply to it all the transformations of the principal group. We thus obtain an infinite manifoldness with a number of dimensions in general equal to the number of arbitrary parameters contained in the group, but reducing in special cases, namely, when the configuration originally selected has the property of being transformed into itself by an infinite number of the transformations of the group. Every manifoldness generated in this way may be called, with reference to the generating group, a body.

This means in modern language, that if GG is the given group acting on a given space XX, and if S↪XS \hookrightarrow X is a given subspace (a configuration), then the “body” (“Körper”) generated by this is the coset

The text goes on to argue that spaces of this form G/StabG(S)G/Stab_G(S) are of fundamental importance:

If now we desire to base our investigations upon the group, selecting at the same time certain definite configurations as space-elements, and if we wish to represent uniformly things which are of like characteristics, we must evidently choose our space-elements in such a way that their manifoldness either is itself a body or can be decomposed into bodies.

Higher Klein geometry

Logicality and invariance

Logicians have attempted to demonstrate that specifically logical constructions are those invariant under the largest group of transformations, in the sense of the Erlangen program. See logicality and invariance.